射影覆盖、代数学说与关系商完备化
Projective covers, doctrines of algebras and the relational quotient completion
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中文总结 AI 辅助
本文研究带商的关系学说中的射影对象，将外延商完备化得到的射影对象刻画为允许射影覆盖的对象，并将结果应用于单子代数学说，推广了正合范畴上单子范畴的相关结论。
中文摘要 AI 辅助
关系学说的外延商完备化，为带弱有限极限范畴的正合完备化与存在性初等学说的初等商完备化提供了统一推广。本文研究带商的关系学说中的射影对象，将通过外延商完备化得到的射影对象刻画为那些允许射影覆盖的对象。我们将该结果应用于带商的关系学说上的单子代数学说，描述这些学说在何种情形下可作为其对自由代数（适当子范畴）限制的外延商完备化。这推广了正合范畴上单子范畴的类似结果，还涵盖了更多例子，如度量空间范畴上的单子，它们产生了量化代数的变体。
英文摘要
The extensional quotient completion of relational doctrines provides a common generalization of both the exact completion of categories with weak finite limits and the elementary quotient completion of existential elementary doctrines. In this paper, we study projective objects in relational doctrines with quotients, characterizing those obtained through the extensional quotient completion as those admitting a projective cover. We apply this result to doctrines of algebras for monads on relational doctrines with quotients, describing in which cases these arise as the extensional quotient completion of their restriction to (appropriate subcategories of) free algebras. This extends a similar result for monadic categories over exact ones, covering also more examples such as monads over the category of metric spaces giving rise to variants of quantitative algebras.
